Neither of those sentences is clear

Agreed. Ditch the ambiguous “her”.

“My mom, grandma, and I went to my grandma’s favorite store.”

How about this: “It was my grandma’s birthday and we wanted to make it special! My mom, my grandma, and I all went to her favorite store.” I’d argue that using “my grandma” instead of “her” would be unpleasantly repetitive there.

That approach is marginally better. It’s still arguably unclear. The store could be your mom’s favorite store to get special gifts from

If you think using Grandma three times is too much, you could replace the first Grandma in the second sentence with “she.” For instance, “she, my mom, and I went…”

My Mom went to her favourite store, and took Gran and me with her.

> My mom took my grandma and I to her favorite store

Shouldn't that be me, not I?